x^2-21x-1=0

Solution for x^2-21x-1=0 equation:

Simplifying
x2 + -21x + -1 = 0

Reorder the terms:
-1 + -21x + x2 = 0

Solving
-1 + -21x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -21x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + -21x + x2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -21x + x2 = 0 + 1
-21x + x2 = 0 + 1

Combine like terms: 0 + 1 = 1
-21x + x2 = 1

The x term is -21x.  Take half its coefficient (-10.5).
Square it (110.25) and add it to both sides.

Add '110.25' to each side of the equation.
-21x + 110.25 + x2 = 1 + 110.25

Reorder the terms:
110.25 + -21x + x2 = 1 + 110.25

Combine like terms: 1 + 110.25 = 111.25
110.25 + -21x + x2 = 111.25

Factor a perfect square on the left side:
(x + -10.5)(x + -10.5) = 111.25

Calculate the square root of the right side: 10.547511555

Break this problem into two subproblems by setting 
(x + -10.5) equal to 10.547511555 and -10.547511555.

Subproblem 1
x + -10.5 = 10.547511555

Simplifying
x + -10.5 = 10.547511555

Reorder the terms:
-10.5 + x = 10.547511555

Solving
-10.5 + x = 10.547511555

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '10.5' to each side of the equation.
-10.5 + 10.5 + x = 10.547511555 + 10.5

Combine like terms: -10.5 + 10.5 = 0.0
0.0 + x = 10.547511555 + 10.5
x = 10.547511555 + 10.5

Combine like terms: 10.547511555 + 10.5 = 21.047511555
x = 21.047511555

Simplifying
x = 21.047511555

Subproblem 2
x + -10.5 = -10.547511555

Simplifying
x + -10.5 = -10.547511555

Reorder the terms:
-10.5 + x = -10.547511555

Solving
-10.5 + x = -10.547511555

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '10.5' to each side of the equation.
-10.5 + 10.5 + x = -10.547511555 + 10.5

Combine like terms: -10.5 + 10.5 = 0.0
0.0 + x = -10.547511555 + 10.5
x = -10.547511555 + 10.5

Combine like terms: -10.547511555 + 10.5 = -0.047511555
x = -0.047511555

Simplifying
x = -0.047511555

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {21.047511555, -0.047511555}